 
Summary: Final Problem Set.
1. Getting caught up.
The following exercises from previous problem sets I consider important. Do
them before you do any others if you are trying to get caught up.
(i) Homework One: Exercise 2.1;
(ii) Homework Three: 3;
(iii) Homework Six: 2.1, 2.2, 4.1, 4.2;
(iv) Homework Seven: Oscillation, 3.1, 3.2;
(v) Homework Eight: 1.1, 1.2, 1.5;
(vi) Homework Nine: all of it.
2. Some topology.
Suppose X and Y are topological spaces and f : X Y .
1. Suppose U is a family of open subsets of X; X U; and fU is continuous on
U for each U U. Show that f is continuous.
2. Suppose F is a finite family of closed subsets of X; XF; and fF is continuous
on F for each F F. Show that f is continuous.
3. Show by counterexample that if F is not finite in 2. then f need not be
continuous.
3. Power series.
Do Exercise 1.4 in the notes on Power Series.
